NAF Conversion: An Efficient Solution for the Range Matching Problem in Packet Filters

1. NAF Conversion: An Efficient Solution for the Range Matching Problem in Packet Filters Author: Nizar Ben Neji, Adel Bouhoula Publisher: IEEE International Conference ,2011 Presenter: Kai-Yang Liu Date:2011/08/31

2. Introduction • The cause of the coexistence of range based and prefix based fields within the filtering rules. • In this paper, we deal with prefix-based solutions. • The available solutions usually require that the filtering rules must be written only in prefix format. • Arbitrary ranges need to be usually converted into standard prefixes when dealing with prefix based solutions.

3. Direct Range to Prefix Conversion

4. Exmaple

5. Direct Range to Prefix Conversion • In the worst case, the w-bit range [1, 2w−2] is split into 2w−2 prefixes. • For example, the 4-bit worst case range [1,14] requires 6 prefixes as follows:

6. Non-Adjacent Form • Every integer n has a unique non-adjacent form (NAF). NAF is a signed binary representation (SBR) of length k and of the form n = where the digits are in {−1, 0, 1} satisfying the property that ai×ai+1= 0 • Example: 47 is 101111 = 32+8+4+2+1 and the NAF form is

7. Proposed Technique • Notations and Terminology: 1. w is the size of the packet header field to be inspected. 2. A w-bit range R = [l , u]satisfies 0 ≤ l ≤ 2w−1 (lower bound) and 0 ≤u ≤ 2w −1 (upper bound) where l ≤ u.

8. Proposed Technique 3. Adjacent ranges: Two w-bit ranges [l1 , u1] and [l2 , u2] are adjacent if l2 = u1+1. 4.Consecutive elementary ranges: Two elementary ranges[l1 , u1] and [l2 , u2] are consecutive if they are adjacent and they have same widths or consecutive power of 2 widths. => |width1−width2| = min(width1 , width2)

9. Conversion Algorithm of a Single Range • Lower{ } is a list of Integers and it stores lower bounds of ranges, • Upper{ } is a list of Integers and it stores upper bounds of ranges, • Sign{ } is a binary list and it stores signs as 0s and 1s,the zero value means “＋” and one means “－”.

10. Example

11. Example • Case 1: difference equals to zero Ex: [4,7] ,[8,11] [4,7] = 01** U1+1=L2 , L2=(1000)2 [8,11] = 10** max(width1,width2)=4 => L2 mod (2.max(midth1,width2))=0 ∴ can’t be converted

12. Example • Case 2: difference equals to width1 Ex: [6,7] ,[8,11] [6,7] = 011* U1+1=L2 , L2=(1000)2 [8,11] = 10** max(width1,width2)=4 => L2 mod (2.max(midth1,width2))=0 ∴ can’t be converted

13. Matching Algorithm • Two main phases: searching for the longest matching prefix and then searching for the shortest prefix that doesn’t match. Ex: A. 2 = (0010)2 B. 4 = (0100)2.

14. Performance Analysis • Worst Expansion Case Theworst expansion case for DRPC algorithm gives for the w-bit range of the form [1, 2w−2] 2w-2 prefixes. The proposed algorithm: 　→ Only 3 prefixes

15. The Worst Case for the NAF Conversion • The worst case expansion is only w prefixes • Example:

16. General Case of the Worst Result

17. Average Expansion Case

18. Special Cases • Converting multiple ranges: 1. Two intersecting ranges [a , b] and [c , d] and c≤b 2.Two disjoint ranges such that b+1<c • Left and right extremal ranges: Left: [0, a ] with 0≤a≤2W-1；Worst:[0, 2W-2] Right: [b, 2W-1 ] with 0≤b≤2W-1；Worst:[1, 2W-1]

19. Special Cases • Negation of an arbitrary range: The negation of the w-bit range [a, b] gives two extremal w-bit ranges [0, a−1] and [b+1, 2w−1]. • Worst case of DRPC : 2w-2 prefixes given by the range of the form • Worst case of NAF conversion : w prefixes given by the negation of the range of the form: